Boundary singularities in mean curvature flow and total curvature of minimal surface boundaries

Abstract

. For hypersurfaces moving by standard mean curvature flow with boundary, we show that if the tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in R 3 that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces. Let κ be the largest number with the following property: if M is a minimal surface in R 3 bounded by a smooth simple closed curve of total curvature < κ , then M is a disk. Examples show that κ < 4 π . In this paper, we use mean curvature flow to show that κ > 3 π . We get a slightly larger lower bound for orientable surfaces.